Calculation of metamorphic two-dimensional quantum

Calculation of metamorphic two-dimensional quantum energy system: Application to wetting layer states in InAs/InGaAs metamorphic quantum dot nanostructures L. Seravalli, G. Trevisi, and P. Frigeri Citation: J. Appl. Phys. 114, 184309 (2013); doi: 10.1063/1.4830021 View online: http://dx.doi.org/10.1063/1.4830021 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i18 Published by the AIP Publishing LLC.

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JOURNAL OF APPLIED PHYSICS 114, 184309 (2013)

Calculation of metamorphic two-dimensional quantum energy system: Application to wetting layer states in InAs/InGaAs metamorphic quantum dot nanostructures L. Seravalli, G. Trevisi, and P. Frigeri IMEM-CNR Institute, Parco Area delle Scienze, 37/A 43100 Parma, Italy

(Received 18 June 2013; accepted 28 October 2013; published online 13 November 2013) In this work, we calculate the two-dimensional quantum energy system of the In(Ga)As wetting layer that arises in InAs/InGaAs/GaAs metamorphic quantum dot structures. Model calculations were carried on the basis of realistic material parameters taking in consideration their dependence on the strain relaxation of the metamorphic buffer; results of the calculations were validated against available literature data. Model results confirmed previous hypothesis on the extrinsic nature of the disappearance of wetting layer emission in metamorphic structures with high In composition. We also show how, by adjusting InGaAs metamorphic buffer parameters, it could be possible: (i) to spatially separate carriers confined in quantum dots from wetting layer carriers, (ii) to create an hybrid 0D-2D system, by tuning quantum dot and wetting layer levels. These results are interesting not only for the engineering of quantum dot structures but also for other applications of metamorphic structures, as the two design parameters of the metamorphic InGaAs buffer (thickness and composition) provide additional degrees of freedom to control properties of interest. C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4830021] V I. INTRODUCTION

In lattice-mismatched structures with layers thicker than the critical thickness for plastic relaxation, the lattice parameter changes from the value of the underlying substrate to the free-standing value of topmost material. From this perspective, these layers, known as metamorphic buffers (MBs), behave as virtual substrates where the lattice parameter can be controlled without acting on the composition of the alloy. The density of dislocations that allow the relaxation of the strain in the layer can be reduced by orders of magnitude by an adroit design of the composition profile of the MB, keeping them separated from the active region of the device. Metamorphic semiconductor structures grown on GaAs substrates have been actively researched for a variety of applications, ranging from triple-junction1 and quantum dot (QD)2 solar cells to light-emitting devices,3,4 from High Electron Mobility Transitors (HEMTs)5 to Semiconductor Optical Amplifiers (SOAs).6 However, despite the relevant success of metamorphic structure, little attention has been paid to consider in depth their basic physical properties and their potentialities in terms of advanced design possibilities. From a theoretical point of view, although transport properties of metamorphic structures have been investigated,7,8 to the best of authors knowledge, no work has been conducted on the effects of strain and composition on the electronic properties of low-dimensional metamorphic structures of interest for applications in photonics. In this theoretical work, we study InAs/InGaAs metamorphic systems, by calculating the wetting layer (WL) states. The results of this work, beyond a relevant interest for QD-based structures, can be considered of general validity for 2D metamorphic systems. 0021-8979/2013/114(18)/184309/8/$30.00

Metamorphic InAs/InGaAs QDs have been proven to be interesting structures and rather different from pseudomorphic InAs/GaAs,9 in particular for redshifting the emitted light in the 1.3–1.55 lm optical window, of great interest for fiber-optic telecommunication devices.10–12 In addition to this technological interest, the metamorphic system has the unique advantage of providing two independent design parameters (composition and thickness) that allow to engineer the quantum system on a wide range: this has been studied in depth for the QD system,13,14 but not considered so far for 2D structures. It is widely known that the thin InAs layer that is present below InAs QDs, termed WL affects substantially QD properties.15–17 This is true both for confined carrier dynamics, as WL states are channels for carrier relaxation,18,19 and for Coulomb interaction effects between QD and WL carriers, in particular for structure where optical emission from single QD is studied.20,21 Hence, an in-depth theoretical study of the metamorphic WL system is justified, to explore the possibilities of this system to control the WL energy levels, both in terms of energy values and of carrier localization. We emphasize that in this work, we do not calculate QD levels that have been already modelled in previous works, as here our interest is devoted to the WL quantum system. We used Tibercad software to calculate InAs WL states grown on InxGa1–xAs partially relaxed meta-buffers. We focused in particular on the effects of composition and thickness of the MBs on 2D energy states and on carrier wavefunctions. As WL parameters such as thickness, composition, and strain are not easily available for metamorphic structures—X-Ray Diffraction (XRD) and Transmission Electron Microscopy (TEM) characterization are difficult due to the presence of defects—model calculation were based on reasonable assumptions that, nevertheless, can give important

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and InyGa1yAs layers, to explore how this more complex design may open the way to new engineering possibilities. The results of our work aim at answering some questions that could give new precious hints to the design of complex quantum structures: (i) is it possible to create hybrid 0D-2D systems using MBs, where QD and WL levels might resonate in energy?, (ii) is it possible to push carriers out of the WL, far from the QDs, to reduce the effects of Coloumb interactions?, and (iii) what is the reason of the recently reported WL "disappearance" in In-rich metamorphic structures: is it design or material related? II. THEORETICAL MODEL

FIG. 1. (Top) Scheme of the structure used for the model calculation, grown on GaAs (001) substrate and composed by a GaAs buffer layer, an InxGa1xAs MB of thickness t, the In(Ga)As WL, the InyGa1yAs capping layer of thickness 20 nm, and the InAlAs additional layers (when considered). The dashed line indicates the extension of the quantum region considered for solving the eigenvalues problem. (Bottom) Profiles of CB and VB along the vertical direction of the structures considered, dotted lines indicate the band profiles due to additional InAlAs layers, while dashed lines refer to confined levels for electrons and holes in the WL.

semi-quantitative indications, as already done in literature.22,23 We considered two growth methods used to obtain metamorphic QDs: the mainstream Stranski-Krastanow (SK) one, which results in structures with an high density of QDs, for which an InAs WL has been considered and the one recently introduced based on the post-growth annealing of sub-critical InAs coverages, for which an InGaAs WL has been considered, as discussed in depth in section “Theoretical Model.” The latter approach allows to obtain very low density of QDs of great interest for the development of single photon sources.24,25 Model results were validated by the comparison with available literature data for WL energy states, also considering structures where InAs has been overgrown with InAlAs

For the calculation of the WL quantum confined levels for the system composed by the GaAs layers, the InxGa1xAs MB, the In(Ga)As WL, the InyGa1yAs cap layer, and the InAlAs layers (when considered), we used the software Tibercad.26 The geometry of the structure (schematically shown in Fig. 1) was divided in different regions; and subsequently, a computational mesh was used to discretize the partial differential equations representing the physical models to be solved. The one dimensional Schroedinger equation was solved with a single-band effective mass model for the conduction band (CB) and with a 6 band-kp model for the valence band on the discretized mesh geometry. Band profiles are shown in Fig. 1. The eigenvalue problem was solved by calculating first the strain condition of the structure relatively to a “virtual” InGaAs material; calculation of material parameters is described below. Then, a quantum region is selected where the Schroedinger equation was solved for electrons and heavy holes by an iterative process with open boundary conditions: eigenvalues and eigenfunctions were derived by values of convergence of solutions. In Table I, we indicate parameters used to carry on the calculation on the Tibercad software. The QD effect on WL levels has been disregarded in this calculation, experimental work recently showed that this is a reasonable hypothesis.27 The lattice parameter and the energy gap of the InGaAs MB depend on its thickness t and composition x, thus a “virtual” material InGaAs has been considered for every t/x couple. The in-plane lattice parameter ax of the InxGa1  xAs MB is related to the in-plane strain e by e ¼ (ax - a0)/a0, where a0 is the InGaAs free standing lattice parameter. The dependence of e on the t and x parameters was derived by following the model of Maree et al.,28 whose predictions were experimentally confirmed in Refs. 29 and 30.

TABLE I. Solver parameters used to carry on calculation with Tibercad software.26 Parameter Eigensolver tolerance Number of eigenstates Model–electrons Model–holes Mesh–units Quantum region Temperature

Description and unit

Value

Numerical eigensolver tolerance used as a convergence criteria (meV) Number of eigenstates to be computed Number of bands for electrons Number of bands for kp model for holes Maximum distance between geometrical points in mesh grid (nm) Extension of quantum region (nm) Temperature of the system (K)

0.1 3 1 (single band) 6 0.1 40 10

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The energy gaps for different values of x and t were derived from the deformation potential theory31 in which the gap E0HH between the minimum of the CB and the maximum of the heavy hole valence band (HH-VB) can be expressed in terms of e as E0 HH ¼ E0 þ dEH  dES =2;

(1)

where dEH ¼ 2ae (C11  C12)/C11 and dES ¼ 2be (C11 þ 2C12)/ C11; a and b are the hydrostatic and the shear deformation potentials and Cij the elastic stiffness constant. The value of the exciton binding energy was set at 10 meV,32 while values of effective masses, deformation potentials, elastic constants, and other relevant parameters for the “virtual” material fed in the Tibercad model were derived from the available literature data, including the bowing effects for different values of x.33 On the basis of the derived lattice parameter of the MB, prior to the calculation of quantum confined levels, a strain calculation of the InxGa1xAs/In(Ga)As/InyGa1yAs (InAlAs) system is executed by the software to consider the correct value of the mismatch f between the MB and the WL, defined as f ¼ (aInAs - ax)/ax, where aInAs is the lattice parameter of InAs. See Table I of supplemental material for values of mismatch, lattice parameter, strain, and energy gap for different InGaAs MBs.34 With respect to the structural parameters of the thin In(Ga)As layer that constitutes the WL, namely thickness d and In composition, it should be noticed that substantial differences were reported when growing it in the SK regime or with the subcritical coverage approach, thus different hypothesis have been put forward in the two cases. A. Modeling of InAs WL in the Stranski-Krastanow regime

In SK structures, the WL was considered as an InAs square well (SQW), a reasonable assumption as discussed in Ref. 23; in Ref. 27, it was shown that WL energy calculated under this approximation deviates form the experimental value less than 10 meV. d was taken as equivalent to the critical thickness h for the transition from twodimensional to three-dimensional growth; in Ref. 35, it was discussed how h depends on the mismatch f. Experimental values of h were considered when available in the literature,36,37 otherwise values extracted from the curve reported in Ref. 35 were used. Values of h range from 1.2 ML (for MBs with high x and/or small t) to 1.4 ML (for MBs with low x and/or large t) and are reported in Table II of supplemental material.34 A second assumption of this calculation is that the In composition of the WL does not vary when x and t are changed, as experimentally demonstrated for QDs grown on InGaAs MBs, due to the agreement between experimental PL emissions of QDs and model calculations where the QD In composition was fixed in the whole range x/t.13 As the lattice parameter of the metamorphic layer on which InAs is deposited depends on both x and t, it was necessary to calculate the tetragonal distortion due to different

mismatch between InGaAs MBs and InAs WLs, using the relation aWL ðperpÞ ¼ aWL ðfreeÞð1  2  eWL  c12=c11Þ;

(2)

where aWL(perp) is the out-of-plane lattice parameter of the InAs layer, aWL (free) is the free-standing lattice parameter of InAs, and eWL is the in-plane strain of the InAs layer, given by (ax - aWL(free))/aWL(free). See Table I of supplemental material for values of quantities for representative x and t.34 B. Modeling of InGaAs WL in the sub-critical regime

In the case of the deposition of subcritical coverages of InAs on MBs, the structural parameters of the WL have been considered differently, due to the experimental evidences of substantial changes from the SK case.38 As a true critical thickness does not exist in this growth regime, d has been derived by considering the deposited InAs sub-critical coverages: experimental data based on XRD characterization of structures with InAs sub-critical coverage of 1.5 ML showed that the WL was 1.0 nm thick with an In composition of 0.30.38 As we showed that the sub-critical deposition needed to obtain QDs on MBs is larger than on GaAs,25 here we assume a linear proportionality between the subcritical coverage and the WL thickness. By considering the tetragonal distortion discussed above, the data of Ref. 25 may be converted in the following WL thicknesses: 1.5 ML ¼> 1 nm for x ¼ 0, 1.55 ML ¼> 1.1 nm for x ¼ 0.15, 1.65 ML ¼> 1.3 nm for x ¼ 0.22, and 1.8 ML ¼> 1.4 nm for x ¼ 0.30. With respect to the WL composition, as it was proved experimentally that in this growth regime the WL is much poorer in In that in the SK case, we considered an InzGa1zAs WL where the change in z is linear with x, i.e., z ¼ 0.30 þ x. This assumption is reasonable if one considers the effects of both QD/substrate intermixing and In segregation in the MBs. As often discussed for InAs WLs grown on GaAs,23,39,40 it is known that InAs/GaAs intermixing takes place when growing QDs structures. This leads to the incorporation of Ga from the substrate into the WL. In the case of MBs, the fraction of Ga in the growth front is lower, then a lower Ga content (i.e., higher In content) in the WL should be expected with respect to the growth on GaAs. Second, In segregation is known to take place during the epitaxial growth of InGaAs alloys,41,42 with the consequence of an In enrichment of the growth front that should increase with the In content in the MB. Although these might be strong assumptions, they should be indeed reasonable and allow to give semi-quantitative indications: more refined calculations could be possible with structural characterization of metamorphic samples to give precise measurements of composition and dimensions of WL. Anyhow, it should be noted that such information is not easily obtained on these kind of structures, where the effect of surface roughness, cross-hatch morphology, and defect density has to be taken into account.

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III. RESULTS A. SK structures

In order to validate the model, we calculated WL energy levels for metamorphic QD structures grown in the SK regime and compared them with available experimental data from the literature. First of all, we considered structures where the QDs are grown on InxGa1xAs MB and capped by an InxGa1xAs layer (symmetric structures); and as a second step, we also considered the comparison with experimental data from asymmetric structures where the QDs are capped by an InyGa1yAs or InyAl1yAs layer. In Figure 2, we show the WL emission energies as measured by PR43 and by Photoluminescence Excitation Spectroscopy (PLE)44 as functions of the mismatch between InAs WL and InxGa1xAs MBs with x ranging from 0.09 to 0.35 (see Table II of supplemental material for additional structural parameters);34 model calculations of WL energy levels for same structures are represented by the solid lines that go from a pseudomorphic layer (with same lattice parameter of GaAs and, hence, f ¼ 7.14%) to a completely relaxed layer (with lattice parameter corresponding to the free-standing one of InGaAs and with the minimum possible value of f). For PR data at temperature different from 10 K, a Varshni shift from RT to 10 K of 63 meV was used. The errors for the model results (reported as vertical bars) stem from uncertainties in transition thickness (that can be estimated in 60.1 ML) and in the In composition of layers (63%).14 The WL emissions are reduced with decreasing mismatch due to two effects: (i) the reduction of the strain in the WL layers that causes a lowering of the InAs energy gap and (ii) the slight increase of the critical thickness (hence, of the WL thickness) that has been discussed in Ref. 35. Experimental works did not show WL emissions in structures with x ¼ 0.35, a feature explained by increase surface roughness that might inhibit WL formation. It can be seen how for almost all data, the agreement between model calculation and experimental data is less than 20 meV, a noteworthy result if one considers the strong starting hypothesis on the absence of change in WL composition for different x/t values. However, it can be noticed that in general the WL PL emissions are at energy values consistently lower than the model calculations, with a larger discrepancy for structures with higher x: this could be an indication that there is a change in structural parameters of WL grown on high-x surfaces, a conclusion reached also in Ref. 35. For structures with x  0.28 and large MB thickness (i.e., low f), it has been reported that no WL levels are detectable by optical techniques.43,44 Present calculations allow to conclude if this effect is due to intrinsic properties of the quantum system, such as band discontinuities between MB and WL too low to confine carriers in the 2D states. In Fig. 3, we report the model calculation for the electron and heavy holes states of the MB and of the WL for x ¼ 0.28, 0.31, and 0.35, showing their dependence on the mismatch f. In the same graph, we indicate with filled squares structures where WL levels have been detected and with open squares structures where no WL levels have been detected.43,44

FIG. 2. Energy of WL transitions as deduced by PR (Ref. 43—circles) and by PLE (Ref. 44—squares) for InAs/InxGa1xAs metamorphic structures as functions of the WL-MB mismatch against model calculation (solid lines). Dashed vertical bars indicate errors in model results due to uncertainties in input parameters.

It can be seen how by increasing x WL states become very close to MB states, in particular for electrons: however, WL levels were not detectable in structures with same x but with larger d (i.e., smaller mismatch). It can be clearly seen that, if x is unchanged, a reduction of the mismatch results in an increase of the energy difference between WL and MB states, due to effect of increase of the WL thickness discussed above. Hence, from a theoretical point of view, there

FIG. 3. Model calculation of WL (dashed line) and MB (solid line) electron (leftside vertical scale) and heavy holes (rightside vertical scale) energy levels for InAs/InxGa1xAs metamorphic structures as functions of the WL-MB mismatch for x ¼ 0.28, 0.31, and 0.35. Open squares indicate structures where no WL states have been detected, while filled squares refer to structures where WL states have been detected.43,44

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FIG. 4. Energy of WL transitions as deduced by PL (Ref. 25)—full circles) against model calculation (open squares) as functions of x. Values of y of upper layers (Ref. 25) are indicated. Inset: 10 K PL spectra for structures with x ¼ 0.22 and x ¼ 0.22 (red), y ¼ 0.30 (blue). The WL value calculated by the model is indicated by dashed lines.

is no intrinsic reason that explains why WL levels are not detected for large thickness: therefore, previous hypothesis on extrinsic factors related to the InGaAs materials, such as rough surfaces, composition inhomogeneity, and higher defect density are reinforced.9,45 In order to further validate the predictability of the model on metamorphic structures, we considered more complex structures where the capping layers is composed by a different material from the MB (asymmetric structures), taking available data from different works appeared in the literature: (i) with 10 nm-thick InyGa1yAs capping layers (y > x)36,37 and (ii) with 6 nm-thick InyAl1yAs layers below and above InAs WL (see Fig. 1 for structure schematics and band profiles).43 In Table III of supplemental materials,34 we report the structure parameters, where it can be noticed how the critical thicknesses for 2D-3D transition reported in the published works in some cases are different from those reported in Ref. 35, conceivably due to different growth conditions. Nevertheless, in both cases, the agreement between model calculation and experimental data is quite good. For structures of case (i), the reported experimental WL energy level for structures with x ¼ 0.16, y ¼ 0.30 was 1.088 eV (Ref. 36) and present calculation give a WL level at 1.090 eV; for x ¼ 0.20, y ¼ 0.40, the experimental data was 1.123 eV (Ref. 37) against a 1.110 eV calculated value; and for x ¼ 0.25, y ¼ 0.45, the experimental data was 1.077 eV (Ref. 37) while the model calculated value was 1.070 eV. In case (ii) for x ¼ 0.15 and y ¼ 0.20 and 0.10, the experimental WL energy was 1.340 eV and a 1.355 eV, respectively,43

comparing with model values of 1.345 eV and 1.350 eV. These results confirm that the model can help predict metamorphic 2D states also for more complex structure design. Incidentally, we note that structures with InAlAs layers are a good example of the usefulness of this model to control the quantum energy system by an adroit design engineering of the metamorphic structure: from these results, it is proven that it is possible to raise WL levels above InxGa1xAs MB energy states by using InAlAs embedding layers below and above QDs, with interesting consequences for QD carrier dynamics.

FIG. 5. Model calculation of WL energy (continuous line) and experimental QD peak emission (open squares) as deduced by PL (Ref. 25) in sub-critical structures as functions of x. Vertical bars indicate FWHM values of QD emission. Dashed line is a guide for the eye.

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FIG. 6. Plots along growth direction of: VB profile (thicker blue line), CB profile (thicker red line), ground electron (red line) and heavy hole (blue line) energy levels, first electron (dashed red line), and heavy hole (dashed blue line) excited levels (leftside vertical scale) calculated by the model. Plots along growth direction of model calculations of squared absolute values of electron (blue line) and heavy hole (red line) wavefunctions (right handside).

B. Sub-critical coverages structures

The interest in these structures relies in the very low density of QDs that is attainable, making them attractive for the development of single photon sources in the 1.3–1.55 lm range. From this perspective, the properties of WL levels and their dependence on MB parameters are of great interest due to the fact that, beside effect on the thermal properties of QD emission, WL-confined carriers levels are known to influence single photon characteristics via Coulomb interaction.21 In Figure 4, we present comparison of model calculation of WL states and experimental PL emission of WL levels as functions of the MB composition x for structures where the MB thickness was fixed to 500 nm (hence with fixed MB-QD mismatch for same x), while InyGa1yAs capping layers with y > x were used to further red-shift the QD emission.25 Due to the low density of QDs, standard PL characterization allows to detect light emission also from WL states, in contrast with SK structures. The agreement between model calculation and PL peaks is always less than 15 meV (except for the x ¼ 0.15 y ¼ 0.45 case). Data are taken from Ref. 25 and from unpublished data for x ¼ 0.15, y > x (see supplemental material for experimental details and for structural parameters).34 For x ¼ 0.30, no emission has been reported from WL, similarly to SK structures, although the model predicts to have WL levels at the 1.030 eV for x ¼ y ¼ 0.30 at about 70 meV from QD peak emission. Previous indications of effect of high-x, high-t surfaces on WL formation that have been discussed in depth in the previous sections are confirmed also for structures deposited by sub-critical coverages. From the comparison of WL levels calculation and PL experimental data, it can be concluded that the model is validated also for structures with a low density of QDs (with discrepancies in the range of 15 meV) and it could be used for advanced metamorphic structure design, for example, to engineer 2D carrier energy and localization.

As a very interesting case of study, let us consider structures with x ¼ 0.22 (whose PL spectra are shown in the inset of Fig. 4), where the expected WL emission falls in the tail of the large QD emission (in low density structures, QD present very wide emission due to higher dishomogeneities of QD size and/or compositions). This suggests that it could be possible to have small energy difference between energy levels for carriers confined in some isolated QDs (emitting in the high-energy tail) and 2D WL levels: this could result in an increase of the electronic coupling between single electrons in QDs and the 2D electron gas in the WL, up to a possible establishment of an hybrid 0D-2D system, a topic of great recent interest.46–48 In order to make some more general assessment on this possibility, let us consider, in Fig. 5, the PL emission of WL and of sub-critical QDs25 presented as functions of x. It can be noticed how the energy difference between QD and WL emission reduces with the increase of the MB In composition. This dependence allows to predict that in the high-x ranges (x  0.25), the WL emission energy difference with QDs could become less than the full width half maxima (FWHM) of the QD PL spectrum (indicated in Figure 5 as a vertical bar of about 100 meV): this means that a good fraction of isolated QDs could be in resonance with WL levels. However, it should be kept in mind that there is the limitation on x related to the absence of WL states discussed above: for this reason, an hybrid 0D-2D systems composed by an isolated QD and the WL could be realized for values of x ranging from 0.20 to 0.30. As a second example of the usefulness of this model to predict properties of metamorphic 2-dimensional quantum system, let us consider the spatial localization of carriers confined in the WL. In Fig. 6, we show the band profiles and the squared values of the wavefunctions of carriers for the metamorphic system with x ¼ 0.15, using capping layers going from y ¼ 0.15 to 0.35, for structures grown by the deposition of subcritical

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FIG. 7. (Left handside) Overlap of ground electron and heavy hole wavefunctions in WL-QW levels as function of y from model calculations (crosses). (Right handside) Probability of finding a carrier within 1 nm of the WL as functions of y from model calculations for heavy holes (circles) and electrons (squares). Lines are guides for the eye.

InAs coverages. In the case of x ¼ 0.25 and 0.35, a mixed WL-QW system emerges, where the spatial localization of carriers may greatly differ. This is a system that has been explored recently, as a possible mean to control the Coulomb interaction between QD and WL carriers via their spatial separation.27 When y ¼ 0.15, carriers in the ground levels are confined within the WL; while in the y ¼ 0.25 case, the electron spreads into the cap layer and the heavy hole has still a greater probability of being in the WL than in the InGaAs capping layer. When y ¼ 0.35, the situation is totally different as both carriers are delocalized in the upper layer and the WL does not seem to have any effect in confining them. This is a very interesting result, as it has been very recently experimentally proved that heavy holes cannot be completely delocalized from the WL in pseudomorphic InAs/InGaAs QD structures.38 In Fig. 7, we present the overlap integral of the heavy hole-electron wavefunctions for the ground states in the WL and the carrier probability of being found in the WL as functions of y in the whole range 0  y  0.35 (x value remains constant at 0.15). It is interesting to notice that while electrons are weakly localized within the WL even for y ¼ 0.15 and spread in the capping layer (for y > x) or in the MB (for y < x), heavy holes become completely delocalized from the WL only when y ¼ 0.35, effectively spatially separating the 2D system from the 0D QD system. Also the overlap integral (proportional to the radiative recombination probability) changes consistently, reducing by almost the half when y ¼ 0 (due to the delocalization of electrons) and by much less when y ¼ 0.35 (due to the concomitant delocalization of electron and heavy holes from the WL to the InGaAs capping layer). The different dependence of 2D carrier localization on the structure design suggests the possibility to engineer the interaction between QD and WL carriers, also by selecting negative or positive 2D charge carriers to electrically interact with QD carriers. This is a very interesting example of how the metamorphic system is a flexible system, with a wider degree of engineering possibilities with respect to the

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pseudomorphic system, for advanced design of structures for single photon sources. As showed above, these possibilities include: (i) engineering of WL carrier spatial localization in structures for single photon emission, to reduce the Coulomb interactions with QD confined carriers, (ii) separation of electrons and heavy holes in the 2D WL system, thus reducing the radiative recombination processes and influencing the QD carrier dynamics, and (iii) find conditions for strong electronic coupling between QD-WL states of great interest for single photon devices. In structures where carriers are electrically driven into QDs, the 0D–2D system could be switched on/off by simple application of external bias. Hence, from these results, it can be concluded that the metamorphic QD system is not only a viable approach to obtain single photon emission in the 1.3–1.55 lm range, but can provide useful tools to control electronic interactions between 0D and 2D carriers. IV. CONCLUSIONS

In conclusion, we developed a simple model for the calculation of quantum 2D states in metamorphic buffers and applied it to the case of InAs WL in metamorphic InAs/InGaAs QD nanostructures. The model was validated against experimental data from different research groups, for structures grown in the Stranski-Krastanow regime and with sub-critical coverages. The analysis of model results highlighted the unique properties of metamorphic system of having two independent design parameters, namely the MB thickness and composition that determine the MB-WL mismatch. The effect of the change of mismatch and of MB composition on WL properties was considered, allowing to reach interesting conclusions. It was confirmed that the previously reported WL disappearance in InAs/InGaAs system for high x/high t is an extrinsic effect, probably related to the surface roughness on which InAs is deposited. For structures with a low density of QDs, the effect of the change in composition of the capping layer on the spatial delocalization of 2D carriers (that influence the Coulomb interaction between QD and WL carriers) and on the electron-heavy hole spatial separation in the QW-WL system (that determines the probability of radiative recombination) was studied and allowed to draw interesting conclusions. In metamorphic InAs/InGaAs system, it could be possible to expel both carriers from the WL, a feature not obtainable in pseudomorphic GaAs-based structures, where only the electron can be delocalized from the WL. Moreover, model calculations suggest that in these systems, it could be possible to have QD and WL levels resonate in energy (at least for a fraction of the QD ensemble). On the basis of these model results, it seems viable to propose the use of metamorphic InAs/InGaAs QDs as: (i) a possible system to have hybrid 2D–0D quantum levels and (ii) a scheme for QD-based single photon sources where the interaction between WL and QD carrier can be engineered. More in general, one could think at the metamorphic approach as a system that allows to engineer two properties of the QW system: this could be a feature not limited to QD structures, but it could also have applications in other metamorphic 2-dimensional structures (QW lasers,

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Seravalli, Trevisi, and Frigeri

high mobility devices, such as MHEMT, semiconductor optical amplifiers, and QW solar cells). 1

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